The orbital degrees of freedom refer to different shapes of the wave functions with degenerate energies. In recent years, optical lattices engineered by interfering laser beams offer new means to explore interacting fermions with orbital degrees of freedom the symmetries of which differ from those found in traditional solids. We show that the orbital hopping pattern alone is sufficient for producing topologically non-trivial band structures. We unveil a topological insulator phase of fermions on a two-leg ladder of double-well lattices, similar to those recently realized in experiments.

Understanding the quantum phases of interacting fermions is a fundamental, chanllenging problem in many-body physics. Broken symmetry phases, such as spin density wave order in antiferromagnetic metal Chromium, or the p-wave superfluid order in liquid Helium 3, have long been known and well understood. Motivated by recent experiments, we find theoretically that an unconventional spin-density wave phase with p-wave orbital symmetry in ultracold Fermi gases of polar molecules and magnetic atoms. It is a kind of magnetic order formed on bonds connecting the lattice sites, and can be viewed as the particle-hole analog of p-wave superconductivity.

In recent years, atomic physics has opened a new frontier for the exploration of strongly correlated many-body systems. Atoms can be cooled to sub-nanokelvin temperatures, trapped in a small volume and placed in artificial crystalline potentials or electromagnetic fields created by lasers. Furthermore, interactions between atoms can be controlled. This enables simulations of electronic materials with more ideal properties than found in nature, and testing or developing theories of condensed matter in a new environment. Novel forms of quantum matter can also be engineered using ultra-cold atoms.

A new class of materials with strong spin-orbit coupling, known as topological insulators (TI), are bulk insulators with edge or surface conduction channels that respect the time-reversal (TR) symmetry. In that sense they are similar to quantum Hall systems, which however are not invariant under TR due to the externally applied magnetic field. The Rashba spin-orbit coupling found in TI materials has a “dynamical” symmetry that can shape incompressible quantum liquids in the presence of strong quantum fluctuations, without an analogue in quantum Hall states. Such quantum liquids can exhibit new and not yet experimentally discovered topological orders with Abelian or non-Abelian fractional statistics.